The breadth of symplectic and Poisson geometry: Festschrift in honor of Alan Weinstein PDF
Preview The breadth of symplectic and Poisson geometry: Festschrift in honor of Alan Weinstein
Progress in Mathematics Volume232 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein The Breadth of Symplectic and Poisson Geometry Festschrift in Honor of Alan Weinstein Jerrold E. Marsden Tudor S. Ratiu Editors Birkha¨user Boston • Basel • Berlin JerroldE.Marsden TudorS.Ratiu CaliforniaInstituteofTechnology EcolePolytechniqueFe´de´raledeLausanne DepartmentofEngineering De´partementdeMathe´matiques andAppliedScience CH-1015Lausanne ControlandDynamicalSystems Switzerland Pasadena,CA91125 U.S.A. AMSSubjectClassifications:53Dxx,17Bxx,22Exx,53Dxx,81Sxx LibraryofCongressCataloging-in-PublicationData ThebreadthofsymplecticandPoissongeometry:festschriftinhonorofAlanWeinstein/ JerroldE.Marsden,TudorS.Ratiu,editors. p.cm.–(Progressinmathematics;v.232) Includesbibliographicalreferencesandindex. ISBN0-8176-3565-3(acid-freepaper) 1.Symplecticgeometry.2.Geometricquantization.3.Poissonmanifolds.I.Weinstein, Alan,1943-II.Marsden,JerroldE.III.Ratiu,TudorS.IV.Progressinmathematics (Boston,Mass.);v.232. QA665.B742004 516.3’.6-dc22 2004046202 ISBN0-8176-3565-3 Printedonacid-freepaper. (cid:1)c2005Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,Rights andPermissions,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsincon- nectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynow knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (JLS/HP) 987654321 SPIN10958261 www.birkhauser.com Margo,Alan,andAshainParisatthelovelyFontainedesQuatrePartiesduMonde. Contents Preface ............................................................ ix AcademicgenealogyofAlanWeinstein ................................. xiii AboutAlanWeinstein................................................ xv StudentsofAlanWeinstein............................................ xv AlanWeinstein’spublications ......................................... xvi Diracstructures,momentummaps,andquasi-Poissonmanifolds HenriqueBursztyn,MariusCrainic ................................... 1 ConstructionofRicci-typeconnectionsbyreductionandinduction MichelCahen,SimoneGutt,LorenzSchwachhöfer....................... 41 Amathematicalmodelforgeomagneticreversals J.J.Duistermaat.................................................. 59 Nonholonomicsystemsviamovingframes:CartanequivalenceandChaplygin Hamiltonization KurtEhlers,JairKoiller,RichardMontgomery,PedroM.Rios............. 75 Thompson’sconjectureforrealsemisimpleLiegroups SamEvens,Jiang-HuaLu .......................................... 121 TheWeinsteinconjectureandtheoremsofnearbyandalmostexistence ViktorL.Ginzburg ................................................ 139 Simplesingularitiesandintegrablehierarchies AlexanderB.Givental,TodorE.Milanov .............................. 173 Momentummapsandmeasure-valuedsolutions(peakons,filaments,and sheets)fortheEPDiffequation DarrylD.Holm,JerroldE.Marsden.................................. 203 viii Contents Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart, Gerstenhaber,andBatalin–Vilkoviskyalgebras JohannesHuebschmann ............................................ 237 Localizationtheoremsbysymplecticcuts LisaJeffrey,MikhailKogan ......................................... 303 RefinementsoftheMorsestratificationofthenormsquareofthemomentmap FrancesKirwan .................................................. 327 Quasi,twisted,andallthat…inPoissongeometryandLiealgebroidtheory YvetteKosmann-Schwarzbach ....................................... 363 Minimalcoadjointorbitsandsymplecticinduction BertramKostant .................................................. 391 Quantizationofpre-quasi-symplecticgroupoidsandtheirHamiltonianspaces CamilleLaurent-Gengoux,PingXu................................... 423 Dualityandtriplestructures KirillC.H.Mackenzie ............................................. 455 Starexponentialfunctionsastwo-valuedelements Y.Maeda,N.Miyazaki,H.Omori,A.Yoshioka .......................... 483 FrommomentummapsanddualpairstosymplecticandPoissongroupoids Charles-MichelMarle ............................................. 493 ConstructionofspectralinvariantsofHamiltonianpathsonclosedsymplectic manifolds Yong-GeunOh.................................................... 525 TheuniversalcoveringandcoveredspacesofasymplecticLiealgebraaction Juan-PabloOrtega,TudorS.Ratiu ................................... 571 Poissonhomotopyalgebra:Anidiosyncraticsurveyofhomotopyalgebraic topicsrelatedtoAlan’sinterests JimStasheff...................................................... 583 DiracsubmanifoldsofJacobimanifolds IzuVaisman...................................................... 603 Quantummapsandautomorphisms SteveZelditch .................................................... 623 Preface Alan Weinstein is one of the top mathematicians in the world working in the area of symplectic and differential geometry. His research on symplectic reduction, La- grangiansubmanifolds,groupoids,applicationstomechanics,andrelatedareashas had a profound influence on the field.This area of research remains active and vi- brant today and this volume is intended to be a reflection of that vigor. In addition to reflecting the vitality of the field, this is a celebratory volume to honor Alan’s 60th birthday. His birthday was also celebrated inAugust, 2003 with a wonderful week-longconferenceheldattheESI:theErwinSchrödingerInternationalInstitute forMathematicalPhysicsinVienna. Alan was born in New York in 1943. He was an undergraduate at MIT and a graduatestudentatUCBerkeley,wherehewasawardedhisPh.D.in1967underthe directionofS.S.Chern.AfterspendingpostdoctoralyearsatIHESnearParis,MIT, andtheUniversityofBonn,hejoinedthefacultyatUCBerkeleyin1969,becoming afullProfessorin1976. Alan has received many honors, including anAlfred P. Sloan Foundation Fel- lowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to the American Academy of Arts and Sciences in 1992, and an honorary degree at the UniversityofUtrechtin2003. AttheESIconference,S.S.Chern,Alan’sadvisor,sentthefollowingwordsto celebratetheoccasion: “IamgladaboutthiscelebrationandIthinkAlanrichlydeservesit.Alan came to me in the early sixties as a graduate student at the University of California at Berkeley.At that time, a prevailing problem in our geometry group,andthegeometrycommunityatlarge,waswhetheronaRiemannian manifold the cut locus and the conjugate locus of a point can be disjoint. Alanimmediatelyshowedthatthiswaspossible.Theresultbecamepartof his Ph.D. thesis, which was published in the Annals of Mathematics. He received his Ph.D. degree in a short period of two years. I introduced him toIHESandtheFrenchmathematicalcommunity.Hestaysclosewiththem andwiththemathematicalideasofCharlesEhresmann.Heisoriginaland x Preface often came up with ingenious ideas.An example is his contribution to the solutionoftheBlaschkeconjecture.Iamveryproudtocounthimasoneof mystudentsandIhopehewillremaininterestedinmathematicsuptomy age,whichisnow91.’’ Alan’stechnicalcontributionsarewideranginganddeep.Asmanyofhisearly papersinhispublicationlistillustrate,hestartedoffinhisthesisandtheyearsim- mediatelyfollowinginpuredifferentialgeometry,atopichehascomebacktofrom timetotimethroughouthiscareer. AlreadystartingwithhispostdocyearsandhisearlycareeratBerkeley,hebecame interestedinsymplecticgeometryandmechanics.Inthisareaherapidlyestablished himselfasoneoftheworld’sauthorities,producingimportantanddeepresultsranging from reduction theory to Lagrangian and Poisson manifolds to studies of periodic orbits in Hamiltonian systems. He also did important work in fluid mechanics and plasmaphysicsandthroughthiswork,heestablishedwarmrelationswiththeBerkeley physicistsAllanKaufmanandRobertLittlejohn. Alan’simportantworkonperiodicorbitsinHamiltoniansystemsledhimeven- tuallytoformulatethe“Weinsteinconjecture,’’namelythatforagivenHamiltonian flow on a symplectic manifold, there must be at least one closed orbit on a regular compactcontacttypelevelsetoftheHamiltonian.AlongwithArnold’sconjecture, theWeinsteinconjecturehasbeenoneofthedrivingforcesinsymplectictopology overthelasttwodecades. Alankeptuphisinterestinsymplecticreductiontheorythroughouthislaterwork. For instance, he laid some important foundation stones in the theory of semidirect product reduction as well as in singular reduction through his work on Satake’s V-manifolds, along with finding important links with singular structures in moduli spaces. Intertwinedwithhisworkonsymplecticgeometryandmechanics,hedidexten- siveworkongeometricPDE,eigenvalues,theSchrödingeroperatorandgeometric quantization. Alan took the point of view of microlocal analysis and phase space structuresinhisworkinthisarea,emphasizingthelinkswithquantummechanics.
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